Geographically Weighted Logistic Regression- Nigeria Functional and Non Functional Water Points

Setting the scene

  • To build an exploratory model to discover factor affecting water point status in Osun State, Nigeria.

  • Study are: Osun State, Nigeria

  • Data Sets:

  • Osun.rds, contains LGAs boundaries of Osun State. It is in sf polygon data frame, and

  • Osun_wp_sf.rds, contains water points within Osun State. It is in sf point data frame.

Model Variables

  • Dependent Variables: Water point status(i.e. functional/non-functional)

Independent Variables:

  • distance_to_primary_road

  • distance_to_secondary_road

  • distance_to_tertiary_road

  • distance_to_city

  • distance_to_town

  • water_point_population

  • local_population_1Km

  • usage_capacity

  • is_urban

  • water_source_clean

Getting Started

Installing R Packages

Using the code chunk, following packages will be installed into R environment

pacman::p_load(sf, spdep, tmap, tidyverse,funModeling,blorr,corrplot,ggpubr,GWmodel, skimr, caret, tidyr)

Data Import

In this class exercise, two data sets will be used.They are:

Importing analytical data

First, we are going to import the analytical data into R environment.

Osun <- read_rds("rds/Osun.rds")
Osun_wp_sf <- read_rds("rds/Osun_wp_sf.rds")
Osun_wp_sf %>%
  freq(input="status")
Warning: The `<scale>` argument of `guides()` cannot be `FALSE`. Use "none" instead as
of ggplot2 3.3.4.
ℹ The deprecated feature was likely used in the funModeling package.
  Please report the issue at <https://github.com/pablo14/funModeling/issues>.

  status frequency percentage cumulative_perc
1   TRUE      2642       55.5            55.5
2  FALSE      2118       44.5           100.0

From the above chart, it can interpreted that there are 2642 observation of “Functional water points” and 2118 observations of “Non-Functional Water points”.

Visualizing the water point data using tmap

tmap_mode("view")
tmap mode set to interactive viewing
tm_shape(Osun)+
tmap_options(check.and.fix= TRUE)+
  tm_polygons(alpha=0.4)+
tm_shape(Osun_wp_sf)+
  tm_dots(col= "status",
          alpha=0.6)+
  tm_view(set.zoom.limits = c(9,12))

Exploratory Data Analysis

Summary Statistics with Skimr

Osun_wp_sf%>%
  skim()
Warning: Couldn't find skimmers for class: sfc_POINT, sfc; No user-defined `sfl`
provided. Falling back to `character`.
Data summary
Name Piped data
Number of rows 4760
Number of columns 75
_______________________
Column type frequency:
character 47
logical 5
numeric 23
________________________
Group variables None

Variable type: character

skim_variable n_missing complete_rate min max empty n_unique whitespace
source 0 1.00 5 44 0 2 0
report_date 0 1.00 22 22 0 42 0
status_id 0 1.00 2 7 0 3 0
water_source_clean 0 1.00 8 22 0 3 0
water_source_category 0 1.00 4 6 0 2 0
water_tech_clean 24 0.99 9 23 0 3 0
water_tech_category 24 0.99 9 15 0 2 0
facility_type 0 1.00 8 8 0 1 0
clean_country_name 0 1.00 7 7 0 1 0
clean_adm1 0 1.00 3 5 0 5 0
clean_adm2 0 1.00 3 14 0 35 0
clean_adm3 4760 0.00 NA NA 0 0 0
clean_adm4 4760 0.00 NA NA 0 0 0
installer 4760 0.00 NA NA 0 0 0
management_clean 1573 0.67 5 37 0 7 0
status_clean 0 1.00 9 32 0 7 0
pay 0 1.00 2 39 0 7 0
fecal_coliform_presence 4760 0.00 NA NA 0 0 0
subjective_quality 0 1.00 18 20 0 4 0
activity_id 4757 0.00 36 36 0 3 0
scheme_id 4760 0.00 NA NA 0 0 0
wpdx_id 0 1.00 12 12 0 4760 0
notes 0 1.00 2 96 0 3502 0
orig_lnk 4757 0.00 84 84 0 1 0
photo_lnk 41 0.99 84 84 0 4719 0
country_id 0 1.00 2 2 0 1 0
data_lnk 0 1.00 79 96 0 2 0
water_point_history 0 1.00 142 834 0 4750 0
clean_country_id 0 1.00 3 3 0 1 0
country_name 0 1.00 7 7 0 1 0
water_source 0 1.00 8 30 0 4 0
water_tech 0 1.00 5 37 0 20 0
adm2 0 1.00 3 14 0 33 0
adm3 4760 0.00 NA NA 0 0 0
management 1573 0.67 5 47 0 7 0
adm1 0 1.00 4 5 0 4 0
New Georeferenced Column 0 1.00 16 35 0 4760 0
lat_lon_deg 0 1.00 13 32 0 4760 0
public_data_source 0 1.00 84 102 0 2 0
converted 0 1.00 53 53 0 1 0
created_timestamp 0 1.00 22 22 0 2 0
updated_timestamp 0 1.00 22 22 0 2 0
Geometry 0 1.00 33 37 0 4760 0
ADM2_EN 0 1.00 3 14 0 30 0
ADM2_PCODE 0 1.00 8 8 0 30 0
ADM1_EN 0 1.00 4 4 0 1 0
ADM1_PCODE 0 1.00 5 5 0 1 0

Variable type: logical

skim_variable n_missing complete_rate mean count
rehab_year 4760 0 NaN :
rehabilitator 4760 0 NaN :
is_urban 0 1 0.39 FAL: 2884, TRU: 1876
latest_record 0 1 1.00 TRU: 4760
status 0 1 0.56 TRU: 2642, FAL: 2118

Variable type: numeric

skim_variable n_missing complete_rate mean sd p0 p25 p50 p75 p100 hist
row_id 0 1.00 68550.48 10216.94 49601.00 66874.75 68244.50 69562.25 471319.00 ▇▁▁▁▁
lat_deg 0 1.00 7.68 0.22 7.06 7.51 7.71 7.88 8.06 ▁▂▇▇▇
lon_deg 0 1.00 4.54 0.21 4.08 4.36 4.56 4.71 5.06 ▃▆▇▇▂
install_year 1144 0.76 2008.63 6.04 1917.00 2006.00 2010.00 2013.00 2015.00 ▁▁▁▁▇
fecal_coliform_value 4760 0.00 NaN NA NA NA NA NA NA
distance_to_primary_road 0 1.00 5021.53 5648.34 0.01 719.36 2972.78 7314.73 26909.86 ▇▂▁▁▁
distance_to_secondary_road 0 1.00 3750.47 3938.63 0.15 460.90 2554.25 5791.94 19559.48 ▇▃▁▁▁
distance_to_tertiary_road 0 1.00 1259.28 1680.04 0.02 121.25 521.77 1834.42 10966.27 ▇▂▁▁▁
distance_to_city 0 1.00 16663.99 10960.82 53.05 7930.75 15030.41 24255.75 47934.34 ▇▇▆▃▁
distance_to_town 0 1.00 16726.59 12452.65 30.00 6876.92 12204.53 27739.46 44020.64 ▇▅▃▃▂
rehab_priority 2654 0.44 489.33 1658.81 0.00 7.00 91.50 376.25 29697.00 ▇▁▁▁▁
water_point_population 4 1.00 513.58 1458.92 0.00 14.00 119.00 433.25 29697.00 ▇▁▁▁▁
local_population_1km 4 1.00 2727.16 4189.46 0.00 176.00 1032.00 3717.00 36118.00 ▇▁▁▁▁
crucialness_score 798 0.83 0.26 0.28 0.00 0.07 0.15 0.35 1.00 ▇▃▁▁▁
pressure_score 798 0.83 1.46 4.16 0.00 0.12 0.41 1.24 93.69 ▇▁▁▁▁
usage_capacity 0 1.00 560.74 338.46 300.00 300.00 300.00 1000.00 1000.00 ▇▁▁▁▅
days_since_report 0 1.00 2692.69 41.92 1483.00 2688.00 2693.00 2700.00 4645.00 ▁▇▁▁▁
staleness_score 0 1.00 42.80 0.58 23.13 42.70 42.79 42.86 62.66 ▁▁▇▁▁
location_id 0 1.00 235865.49 6657.60 23741.00 230638.75 236199.50 240061.25 267454.00 ▁▁▁▁▇
cluster_size 0 1.00 1.05 0.25 1.00 1.00 1.00 1.00 4.00 ▇▁▁▁▁
lat_deg_original 4760 0.00 NaN NA NA NA NA NA NA
lon_deg_original 4760 0.00 NaN NA NA NA NA NA NA
count 0 1.00 1.00 0.00 1.00 1.00 1.00 1.00 1.00 ▁▁▇▁▁
Osun_wp_sf_clean <- Osun_wp_sf%>%
  filter_at(vars(status,
                 distance_to_primary_road,
                 distance_to_secondary_road,
                 distance_to_tertiary_road,
                 distance_to_city,
                 distance_to_town,
                 water_point_population,
                 local_population_1km,
                 usage_capacity,
                 is_urban,
                 water_source_clean),
            all_vars(!is.na(.)))%>%
  mutate(usage_capacity = as.factor(usage_capacity))

After the above code chunk run, it can be observed 4 observations are deleted and now there are total of 4756 observations with 75 columns.

Learnings from the above code chunk are:

  • exclude missing values (filtering for all_vars(!is.na(.))); and

  • recode usage_capacity as factor (it only has 3 classes) instead of numerical data type. This is because the calibration of logit function will be different.

Correlation Analysis

Using the code chunk below, selected row will be filtered from” Osun_wp_sf_clean” data set and geometry column is dropped.

Osun_wp <- Osun_wp_sf_clean%>%
  select(c(7,35:39,42,43,46,47,57))%>%
  st_set_geometry(NULL)

Next, we plot the correlation matrix for all the numerical data fields.

cluster_vars.cor= cor(
  Osun_wp[,2:7])
corrplot.mixed(cluster_vars.cor,
               lower= "ellipse",
               upper= "number",
               tl.pos= "lt",
               diag= "l",
               tl.col= "black")

From the above result, it can observed there are none of the variables that are highly correlated, i.e. correlation greater than +/- 0.8. Therefore, we will consider all the variables for the further analysis.

Building a Logistic Regression Models

Using the code chunk below, regression model is built.

model <- glm(status ~ distance_to_primary_road+
               distance_to_secondary_road+
               distance_to_tertiary_road+
               distance_to_city+
               distance_to_town+
               is_urban+
               usage_capacity+
               water_source_clean+
               water_point_population+
              local_population_1km,
             data= Osun_wp_sf_clean,
             family= binomial(link= "logit"))

Instead of using typical R report, blr_regress() of blorr package is used.

blr_regress(model)
                             Model Overview                              
------------------------------------------------------------------------
Data Set    Resp Var    Obs.    Df. Model    Df. Residual    Convergence 
------------------------------------------------------------------------
  data       status     4756      4755           4744           TRUE     
------------------------------------------------------------------------

                    Response Summary                     
--------------------------------------------------------
Outcome        Frequency        Outcome        Frequency 
--------------------------------------------------------
   0             2114              1             2642    
--------------------------------------------------------

                                 Maximum Likelihood Estimates                                   
-----------------------------------------------------------------------------------------------
               Parameter                    DF    Estimate    Std. Error    z value     Pr(>|z|) 
-----------------------------------------------------------------------------------------------
              (Intercept)                   1      0.3887        0.1124      3.4588       5e-04 
        distance_to_primary_road            1      0.0000        0.0000     -0.7153      0.4744 
       distance_to_secondary_road           1      0.0000        0.0000     -0.5530      0.5802 
       distance_to_tertiary_road            1      1e-04         0.0000      4.6708      0.0000 
            distance_to_city                1      0.0000        0.0000     -4.7574      0.0000 
            distance_to_town                1      0.0000        0.0000     -4.9170      0.0000 
              is_urbanTRUE                  1     -0.2971        0.0819     -3.6294       3e-04 
           usage_capacity1000               1     -0.6230        0.0697     -8.9366      0.0000 
water_source_cleanProtected Shallow Well    1      0.5040        0.0857      5.8783      0.0000 
   water_source_cleanProtected Spring       1      1.2882        0.4388      2.9359      0.0033 
         water_point_population             1      -5e-04        0.0000    -11.3686      0.0000 
          local_population_1km              1      3e-04         0.0000     19.2953      0.0000 
-----------------------------------------------------------------------------------------------

 Association of Predicted Probabilities and Observed Responses  
---------------------------------------------------------------
% Concordant          0.7347          Somers' D        0.4693   
% Discordant          0.2653          Gamma            0.4693   
% Tied                0.0000          Tau-a            0.2318   
Pairs                5585188          c                0.7347   
---------------------------------------------------------------


Observations

It can be observed first two observations are more than the significance level of 0.05. Therefore, these variables will be excluded for further analysis as they are not significant.

In estimate column, if estimate is positive then that independent variable has positive correlation with dependent variable and if estimate is negative then that independent variable has negative correlation with dependent variable.

In the code chuck below, blr_confusion_matrix() of blorr package is used to compute the confusion matrix of the estimated outcomes by using 0.5 as the cutoff value.

blr_confusion_matrix(model, cutoff= 0.5)
Confusion Matrix and Statistics 

          Reference
Prediction FALSE TRUE
         0  1301  738
         1   813 1904

                Accuracy : 0.6739 
     No Information Rate : 0.4445 

                   Kappa : 0.3373 

McNemars's Test P-Value  : 0.0602 

             Sensitivity : 0.7207 
             Specificity : 0.6154 
          Pos Pred Value : 0.7008 
          Neg Pred Value : 0.6381 
              Prevalence : 0.5555 
          Detection Rate : 0.4003 
    Detection Prevalence : 0.5713 
       Balanced Accuracy : 0.6680 
               Precision : 0.7008 
                  Recall : 0.7207 

        'Positive' Class : 1

The validity of a cutoff is measured using sensitivity, specificity and accuracy.

  1. Sensitivity: The % of correctly classified events out of all events= TP/(TP+FN)
  2. Specificity: The % of correctly classified non-events out of all events= TN/(TN+FP)
  3. Accuracy: The % of correctly classified observation over all observations= (TP+TN)/ (TP+FP+FN+TN)

Observations

From the output, we see that the model gives us an accuracy of 0.668, which is a good start as it is better than guessing (0.5).

The sensitivity and specificity are 0.7207 and 0.6154 respectively. This shows that the true positives (functional water points) are slightly higher than the true negative prediction rates (non-functional water points).

Building Fixed Bandwidth GWR Model

Converting sf data frame to sp data frame

Next, we need to convert the sf data frame into spatial point data frame for GWR model building. This is done using the code chunk below.

Osun_wp_sp <- Osun_wp_sf_clean%>%
  select(c(status,
           distance_to_primary_road,
           distance_to_secondary_road,
           distance_to_tertiary_road,
           distance_to_city,
           distance_to_town,
           water_point_population,
           local_population_1km,
           is_urban,
           usage_capacity,
           water_source_clean))%>%
  as_Spatial()
Osun_wp_sp
class       : SpatialPointsDataFrame 
features    : 4756 
extent      : 182502.4, 290751, 340054.1, 450905.3  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=4 +lon_0=8.5 +k=0.99975 +x_0=670553.98 +y_0=0 +a=6378249.145 +rf=293.465 +towgs84=-92,-93,122,0,0,0,0 +units=m +no_defs 
variables   : 11
names       : status, distance_to_primary_road, distance_to_secondary_road, distance_to_tertiary_road, distance_to_city, distance_to_town, water_point_population, local_population_1km, is_urban, usage_capacity, water_source_clean 
min values  :      0,        0.014461356813335,          0.152195902540837,         0.017815121653488, 53.0461399623541, 30.0019777713073,                      0,                    0,        0,           1000,           Borehole 
max values  :      1,         26909.8616132094,           19559.4793799085,          10966.2705628969,  47934.343603562, 44020.6393368124,                  29697,                36118,        1,            300,   Protected Spring 

Note: We used cleaned version of data set for consistency in the geometrics with our model building (4 water points with missing values excluded).

Computing Fixed Bandwidth

bw.fixed <- bw.ggwr(status ~
                      distance_to_primary_road+
                      distance_to_secondary_road+
                      distance_to_tertiary_road+
                      distance_to_city+
                      distance_to_town+
                      water_point_population+
                      local_population_1km+
                      is_urban+
                      usage_capacity+
                      water_source_clean,
                    data= Osun_wp_sp,
                    family= "binomial",
                    approach= "AIC",
                    kernel= "gaussian",
                    adaptive= FALSE,
                    longlat= FALSE)
bw.fixed
gwlr.fixed <- ggwr.basic(status ~
                      distance_to_primary_road+
                      distance_to_secondary_road+
                      distance_to_tertiary_road+
                      distance_to_city+
                      distance_to_town+
                      water_point_population+
                      local_population_1km+
                      is_urban+
                      usage_capacity+
                      water_source_clean,
                    data= Osun_wp_sp,
                    bw= 2597.255,
                    family= "binomial",
                    kernel= "gaussian",
                    adaptive= FALSE,
                    longlat= FALSE)
Warning in proj4string(data): CRS object has comment, which is lost in output; in tests, see
https://cran.r-project.org/web/packages/sp/vignettes/CRS_warnings.html
Warning in proj4string(regression.points): CRS object has comment, which is lost in output; in tests, see
https://cran.r-project.org/web/packages/sp/vignettes/CRS_warnings.html
 Iteration    Log-Likelihood
=========================
       0        -1957 
       1        -1675 
       2        -1525 
       3        -1441 
       4        -1403 
       5        -1403 

We look at the results below. Similar to when we build multiple linear regression model, the report has 2 sections - generalised regression (global model) results and geographically weighted (GW) regression results. Note that the global model does not have AICc result, so AIC should be used to compare the 2 models.

gwlr.fixed
   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2022-12-18 00:53:19 
   Call:
   ggwr.basic(formula = status ~ distance_to_primary_road + distance_to_secondary_road + 
    distance_to_tertiary_road + distance_to_city + distance_to_town + 
    water_point_population + local_population_1km + is_urban + 
    usage_capacity + water_source_clean, data = Osun_wp_sp, bw = 2597.255, 
    family = "binomial", kernel = "gaussian", adaptive = FALSE, 
    longlat = FALSE)

   Dependent (y) variable:  status
   Independent variables:  distance_to_primary_road distance_to_secondary_road distance_to_tertiary_road distance_to_city distance_to_town water_point_population local_population_1km is_urban usage_capacity water_source_clean
   Number of data points: 4756
   Used family: binomial
   ***********************************************************************
   *              Results of Generalized linear Regression               *
   ***********************************************************************

Call:
NULL

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-124.555    -1.755     1.072     1.742    34.333  

Coefficients:
                                           Estimate Std. Error z value Pr(>|z|)
Intercept                                 3.887e-01  1.124e-01   3.459 0.000543
distance_to_primary_road                 -4.642e-06  6.490e-06  -0.715 0.474422
distance_to_secondary_road               -5.143e-06  9.299e-06  -0.553 0.580230
distance_to_tertiary_road                 9.683e-05  2.073e-05   4.671 3.00e-06
distance_to_city                         -1.686e-05  3.544e-06  -4.757 1.96e-06
distance_to_town                         -1.480e-05  3.009e-06  -4.917 8.79e-07
water_point_population                   -5.097e-04  4.484e-05 -11.369  < 2e-16
local_population_1km                      3.451e-04  1.788e-05  19.295  < 2e-16
is_urbanTRUE                             -2.971e-01  8.185e-02  -3.629 0.000284
usage_capacity1000                       -6.230e-01  6.972e-02  -8.937  < 2e-16
water_source_cleanProtected Shallow Well  5.040e-01  8.574e-02   5.878 4.14e-09
water_source_cleanProtected Spring        1.288e+00  4.388e-01   2.936 0.003325
                                            
Intercept                                ***
distance_to_primary_road                    
distance_to_secondary_road                  
distance_to_tertiary_road                ***
distance_to_city                         ***
distance_to_town                         ***
water_point_population                   ***
local_population_1km                     ***
is_urbanTRUE                             ***
usage_capacity1000                       ***
water_source_cleanProtected Shallow Well ***
water_source_cleanProtected Spring       ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 6534.5  on 4755  degrees of freedom
Residual deviance: 5688.0  on 4744  degrees of freedom
AIC: 5712

Number of Fisher Scoring iterations: 5


 AICc:  5712.099
 Pseudo R-square value:  0.1295351
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian 
   Fixed bandwidth: 2597.255 
   Regression points: the same locations as observations are used.
   Distance metric: A distance matrix is specified for this model calibration.

   ************Summary of Generalized GWR coefficient estimates:**********
                                                   Min.     1st Qu.      Median
   Intercept                                -8.9630e+02 -4.9805e+00  1.7599e+00
   distance_to_primary_road                 -1.9477e-02 -4.8092e-04  3.0174e-05
   distance_to_secondary_road               -1.5757e-02 -3.7583e-04  1.2438e-04
   distance_to_tertiary_road                -1.5673e-02 -4.2538e-04  7.6217e-05
   distance_to_city                         -1.8447e-02 -5.6287e-04 -1.2745e-04
   distance_to_town                         -2.2450e-02 -5.7335e-04 -1.5218e-04
   water_point_population                   -5.2830e-02 -2.2810e-03 -9.8829e-04
   local_population_1km                     -1.2757e-01  5.0016e-04  1.0647e-03
   is_urbanTRUE                             -1.9866e+02 -4.3054e+00 -1.6908e+00
   usage_capacity1000                       -2.0846e+01 -9.7311e-01 -4.1596e-01
   water_source_cleanProtected.Shallow.Well -2.0782e+01 -4.5536e-01  5.3278e-01
   water_source_cleanProtected.Spring       -5.2495e+02 -5.5983e+00  2.5500e+00
                                                3rd Qu.      Max.
   Intercept                                 1.2829e+01 1075.4234
   distance_to_primary_road                  4.8497e-04    0.0143
   distance_to_secondary_road                6.0665e-04    0.0259
   distance_to_tertiary_road                 6.7104e-04    0.0129
   distance_to_city                          2.3763e-04    0.0155
   distance_to_town                          1.9318e-04    0.0225
   water_point_population                    5.0564e-04    0.1313
   local_population_1km                      1.8177e-03    0.0392
   is_urbanTRUE                              1.2864e+00  746.9498
   usage_capacity1000                        3.0334e-01    5.9492
   water_source_cleanProtected.Shallow.Well  1.7870e+00   67.5549
   water_source_cleanProtected.Spring        6.7736e+00  331.1243
   ************************Diagnostic information*************************
   Number of data points: 4756 
   GW Deviance: 2792.323 
   AIC : 4413.603 
   AICc : 4747.217 
   Pseudo R-square value:  0.5726785 

   ***********************************************************************
   Program stops at: 2022-12-18 00:54:08 

Comparing the AIC values of the 2 models, we see that it is lower for the GW regression model at 4,413.603 then for the global regression model at 5,712.09

Model Assessment

Converting SDF into sf data.frame

To assess the performance of the gwLR, firstly, we will convert the SDF object in as data frame by using the code chunk below.

gwr.fixed <- as.data.frame(gwlr.fixed$SDF)

Next, we will label that values greater or equal to 0.5 into 1, else 0. The result the logic comparison operation will be saved into a field called most.

gwr.fixed <- gwr.fixed %>%
  mutate(most= ifelse(
    gwr.fixed$yhat >= 0.5, T, F))

Confusion Matrix

Next, we use confusionMatrix() of caret to display the confusion matrix of the GW model using fixed bandwidth method.

gwr.fixed$y <- as.factor(gwr.fixed$y)
gwr.fixed$most <- as.factor(gwr.fixed$most)
CM <- confusionMatrix(data=gwr.fixed$most, reference= gwr.fixed$y)
CM
Confusion Matrix and Statistics

          Reference
Prediction FALSE TRUE
     FALSE  1824  263
     TRUE    290 2379
                                          
               Accuracy : 0.8837          
                 95% CI : (0.8743, 0.8927)
    No Information Rate : 0.5555          
    P-Value [Acc > NIR] : <2e-16          
                                          
                  Kappa : 0.7642          
                                          
 Mcnemar's Test P-Value : 0.2689          
                                          
            Sensitivity : 0.8628          
            Specificity : 0.9005          
         Pos Pred Value : 0.8740          
         Neg Pred Value : 0.8913          
             Prevalence : 0.4445          
         Detection Rate : 0.3835          
   Detection Prevalence : 0.4388          
      Balanced Accuracy : 0.8816          
                                          
       'Positive' Class : FALSE           
                                          

We see that the accuracy (0.8816 vs 0.66), sensitivity (0.986 vs 0.7207) and specificity (0.9005 vs 0.6154) values have all improved from the non-gwLR global model. By using the gwLR model, we can explain the functional and non-functional water points better now which allows better management of water points through localised strategies (e.g. look at the local neighbourhood regions within Osun state).

Visualizing gwLR

Before we visualise the results of the gwLR model, we clean up the data set for plotting by selecting the relevant data fields (mainly the status column which is the dependent or predicted variable) into a new sf data frame object wp_sf_select in the code chunk below.

wp_sf_select <- Osun_wp_sf_clean %>%
  select(c(ADM2_EN, ADM2_PCODE,
           ADM1_EN, ADM1_PCODE,
           status))

We then combine it with gwr.fixed which has the predicted values of the water point status, in the form of probabilities between 0 and 1.

gwr_sf.fixed <- cbind(wp_sf_select, gwr.fixed)

The code chunk below is used to create an interactive point symbol map.

tmap_mode("view")
tmap mode set to interactive viewing
actual <- tm_shape(Osun) +
  tmap_options(check.and.fix = TRUE) +
  tm_polygons(alpha = 0.4) +
  tm_shape(gwr_sf.fixed) +
  tm_dots(col = "status",
          alpha = 0.6,
          palette = "YlOrRd") +
  tm_view(set.zoom.limits = c(9, 12))

prob_T <- tm_shape(Osun) +
  tm_polygons(alpha = 0.4) +
  tm_shape(gwr_sf.fixed) + 
  tm_dots(col = "yhat",
          border.col = "gray60",
          border.lwd = 1) +
  tm_view(set.zoom.limits = c(9, 12))

tmap_arrange(actual, prob_T, 
             asp = 1, ncol = 2, sync = TRUE)

We see that the predictions are largely aligned with the actual status of the water points, in line with the 88% accuracy rate.

Regression Model after excluding the variables that are not significant

Using the below code chunk, “distance_to_primary_road” and “distance_to_secondary_road” are excluded.

modelM <- glm(status ~ 
                distance_to_tertiary_road+
               distance_to_city+
               distance_to_town+
               is_urban+
               usage_capacity+
               water_source_clean+
               water_point_population+
              local_population_1km,
             data= Osun_wp_sf_clean,
             family= binomial(link= "logit"))
blr_regress(modelM)
                             Model Overview                              
------------------------------------------------------------------------
Data Set    Resp Var    Obs.    Df. Model    Df. Residual    Convergence 
------------------------------------------------------------------------
  data       status     4756      4755           4746           TRUE     
------------------------------------------------------------------------

                    Response Summary                     
--------------------------------------------------------
Outcome        Frequency        Outcome        Frequency 
--------------------------------------------------------
   0             2114              1             2642    
--------------------------------------------------------

                                 Maximum Likelihood Estimates                                   
-----------------------------------------------------------------------------------------------
               Parameter                    DF    Estimate    Std. Error    z value     Pr(>|z|) 
-----------------------------------------------------------------------------------------------
              (Intercept)                   1      0.3540        0.1055      3.3541       8e-04 
       distance_to_tertiary_road            1      1e-04         0.0000      4.9096      0.0000 
            distance_to_city                1      0.0000        0.0000     -5.2022      0.0000 
            distance_to_town                1      0.0000        0.0000     -5.4660      0.0000 
              is_urbanTRUE                  1     -0.2667        0.0747     -3.5690       4e-04 
           usage_capacity1000               1     -0.6206        0.0697     -8.9081      0.0000 
water_source_cleanProtected Shallow Well    1      0.4947        0.0850      5.8228      0.0000 
   water_source_cleanProtected Spring       1      1.2790        0.4384      2.9174      0.0035 
         water_point_population             1      -5e-04        0.0000    -11.3902      0.0000 
          local_population_1km              1      3e-04         0.0000     19.4069      0.0000 
-----------------------------------------------------------------------------------------------

 Association of Predicted Probabilities and Observed Responses  
---------------------------------------------------------------
% Concordant          0.7349          Somers' D        0.4697   
% Discordant          0.2651          Gamma            0.4697   
% Tied                0.0000          Tau-a            0.2320   
Pairs                5585188          c                0.7349   
---------------------------------------------------------------
blr_confusion_matrix(modelM, cutoff= 0.5)
Confusion Matrix and Statistics 

          Reference
Prediction FALSE TRUE
         0  1300  743
         1   814 1899

                Accuracy : 0.6726 
     No Information Rate : 0.4445 

                   Kappa : 0.3348 

McNemars's Test P-Value  : 0.0761 

             Sensitivity : 0.7188 
             Specificity : 0.6149 
          Pos Pred Value : 0.7000 
          Neg Pred Value : 0.6363 
              Prevalence : 0.5555 
          Detection Rate : 0.3993 
    Detection Prevalence : 0.5704 
       Balanced Accuracy : 0.6669 
               Precision : 0.7000 
                  Recall : 0.7188 

        'Positive' Class : 1

It can be observed that there is not much change in the specificity, sensitivity and accuracy rate.

Determining the fixed bandwidth

bw.fixed_M <- bw.ggwr(status ~ distance_to_tertiary_road +
                      distance_to_city +
                      distance_to_town +
                      is_urban +
                      usage_capacity +
                      water_source_clean +
                      water_point_population +
                      local_population_1km,
                      data = Osun_wp_sp,
                    family = "binomial",
                    approach  = "AIC",
                    kernel = "gaussian",
                    adaptive = FALSE, # for fixed bandwidth
                    longlat = FALSE) # input data have been converted to projected CRS
(bw.fixed_M)

Model Assessment

gwlr.fixedM <- ggwr.basic(status ~
                      distance_to_tertiary_road+
                      distance_to_city+
                      distance_to_town+
                      water_point_population+
                      local_population_1km+
                      is_urban+
                      usage_capacity+
                      water_source_clean,
                    data= Osun_wp_sp,
                    bw= 2597.255,
                    family= "binomial",
                    kernel= "gaussian",
                    adaptive= FALSE,
                    longlat= FALSE)
Warning in proj4string(data): CRS object has comment, which is lost in output; in tests, see
https://cran.r-project.org/web/packages/sp/vignettes/CRS_warnings.html
Warning in proj4string(regression.points): CRS object has comment, which is lost in output; in tests, see
https://cran.r-project.org/web/packages/sp/vignettes/CRS_warnings.html
 Iteration    Log-Likelihood
=========================
       0        -2034 
       1        -1772 
       2        -1635 
       3        -1561 
       4        -1530 
       5        -1530 
gwlr.fixedM
   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2022-12-18 00:54:10 
   Call:
   ggwr.basic(formula = status ~ distance_to_tertiary_road + distance_to_city + 
    distance_to_town + water_point_population + local_population_1km + 
    is_urban + usage_capacity + water_source_clean, data = Osun_wp_sp, 
    bw = 2597.255, family = "binomial", kernel = "gaussian", 
    adaptive = FALSE, longlat = FALSE)

   Dependent (y) variable:  status
   Independent variables:  distance_to_tertiary_road distance_to_city distance_to_town water_point_population local_population_1km is_urban usage_capacity water_source_clean
   Number of data points: 4756
   Used family: binomial
   ***********************************************************************
   *              Results of Generalized linear Regression               *
   ***********************************************************************

Call:
NULL

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-129.368    -1.750     1.074     1.742    34.126  

Coefficients:
                                           Estimate Std. Error z value Pr(>|z|)
Intercept                                 3.540e-01  1.055e-01   3.354 0.000796
distance_to_tertiary_road                 1.001e-04  2.040e-05   4.910 9.13e-07
distance_to_city                         -1.764e-05  3.391e-06  -5.202 1.97e-07
distance_to_town                         -1.544e-05  2.825e-06  -5.466 4.60e-08
water_point_population                   -5.098e-04  4.476e-05 -11.390  < 2e-16
local_population_1km                      3.452e-04  1.779e-05  19.407  < 2e-16
is_urbanTRUE                             -2.667e-01  7.474e-02  -3.569 0.000358
usage_capacity1000                       -6.206e-01  6.966e-02  -8.908  < 2e-16
water_source_cleanProtected Shallow Well  4.947e-01  8.496e-02   5.823 5.79e-09
water_source_cleanProtected Spring        1.279e+00  4.384e-01   2.917 0.003530
                                            
Intercept                                ***
distance_to_tertiary_road                ***
distance_to_city                         ***
distance_to_town                         ***
water_point_population                   ***
local_population_1km                     ***
is_urbanTRUE                             ***
usage_capacity1000                       ***
water_source_cleanProtected Shallow Well ***
water_source_cleanProtected Spring       ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 6534.5  on 4755  degrees of freedom
Residual deviance: 5688.9  on 4746  degrees of freedom
AIC: 5708.9

Number of Fisher Scoring iterations: 5


 AICc:  5708.923
 Pseudo R-square value:  0.129406
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian 
   Fixed bandwidth: 2597.255 
   Regression points: the same locations as observations are used.
   Distance metric: A distance matrix is specified for this model calibration.

   ************Summary of Generalized GWR coefficient estimates:**********
                                                   Min.     1st Qu.      Median
   Intercept                                -2.7771e+02 -3.9915e+00  2.9346e+00
   distance_to_tertiary_road                -2.0066e-02 -3.6016e-04  8.9385e-05
   distance_to_city                         -3.0931e-02 -5.6273e-04 -1.0359e-04
   distance_to_town                         -3.4702e-03 -4.3133e-04 -1.2398e-04
   water_point_population                   -3.5450e-02 -2.0856e-03 -1.1271e-03
   local_population_1km                     -5.8060e-02  4.0342e-04  1.0001e-03
   is_urbanTRUE                             -3.0233e+02 -3.1725e+00 -1.4861e+00
   usage_capacity1000                       -4.5295e+01 -1.0249e+00 -3.8880e-01
   water_source_cleanProtected.Shallow.Well -1.0470e+02 -4.2423e-01  5.9626e-01
   water_source_cleanProtected.Spring       -7.9160e+02 -5.4086e+00  2.5525e+00
                                                3rd Qu.      Max.
   Intercept                                 1.0668e+01 1102.7459
   distance_to_tertiary_road                 5.3918e-04    0.0140
   distance_to_city                          1.2672e-04    0.0129
   distance_to_town                          2.2159e-04    0.0161
   water_point_population                    1.9400e-04    0.0569
   local_population_1km                      1.6838e-03    0.0293
   is_urbanTRUE                              8.9541e-01  739.6369
   usage_capacity1000                        3.5031e-01    5.9152
   water_source_cleanProtected.Shallow.Well  1.8040e+00   52.4657
   water_source_cleanProtected.Spring        6.5117e+00  152.2614
   ************************Diagnostic information*************************
   Number of data points: 4756 
   GW Deviance: 3051.369 
   AIC : 4499.24 
   AICc : 4759.621 
   Pseudo R-square value:  0.5330355 

   ***********************************************************************
   Program stops at: 2022-12-18 00:54:46 

We see that both gwLR models have lower AIC values than their global model counter parts.

Converting SDF into sf data frame

gwr.fixed_refined <- as.data.frame(gwlr.fixedM$SDF)
gwr.fixed_refined <- gwr.fixed_refined %>%
  mutate(most = ifelse(
    gwr.fixed_refined$yhat >= 0.5, T, F))

We similarly call the confusion matrix and statistics using confusionMatrix() of caret in the code chunk below.

gwr.fixed_refined$y <- as.factor(gwr.fixed_refined$y)
gwr.fixed_refined$most <- as.factor(gwr.fixed_refined$most)
CM_refined <- confusionMatrix(data = gwr.fixed_refined$most,
                      reference = gwr.fixed_refined$y,
                      positive = "TRUE")
CM_refined
Confusion Matrix and Statistics

          Reference
Prediction FALSE TRUE
     FALSE  1792  302
     TRUE    322 2340
                                          
               Accuracy : 0.8688          
                 95% CI : (0.8589, 0.8783)
    No Information Rate : 0.5555          
    P-Value [Acc > NIR] : <2e-16          
                                          
                  Kappa : 0.7341          
                                          
 Mcnemar's Test P-Value : 0.4469          
                                          
            Sensitivity : 0.8857          
            Specificity : 0.8477          
         Pos Pred Value : 0.8790          
         Neg Pred Value : 0.8558          
             Prevalence : 0.5555          
         Detection Rate : 0.4920          
   Detection Prevalence : 0.5597          
      Balanced Accuracy : 0.8667          
                                          
       'Positive' Class : TRUE            
                                          

We see that the accuracy 0.866, sensitivity 0.88 and specificity 0.84) values have all improved from the non-gwLR global model. By using the gwLR model, we can explain the non-functional water points better now which allows better management of water points through localised strategies (e.g. look at the local neighbourhood regions within Osun state).

The performance measures of the 4 logistic regression models are summarised in the table below.

Performance Measure Global regression with 10 variables gwLR with 10 variables Global regression with 8 variables gwLR with 8 variables
Accuracy 0.6739 0.8837 0.6726 0.8846
Sensitivity 0.7207 0.9005 0.7188 0.8986
Specificity 0.6154 0.8628 0.6149 0.8671

We see that the model accuracy and specificity improve very slightly by removing the non-statistically significant variables from the gwLR model, but the sensitivity drops slightly. Nevertheless, as we would be more interested in finding non-functional water points for maintenance etc., the gwLR model with 8 variables would be more useful with a higher specificity.

Visualizing using tmap

gwr_sf.fixed_refined <- cbind(wp_sf_select, gwr.fixed_refined)
tmap_mode("view")
tmap mode set to interactive viewing
actual <- tm_shape(Osun) +
  tmap_options(check.and.fix = TRUE) +
  tm_polygons(alpha = 0.4) +
  tm_shape(Osun_wp_sf) +
  tm_dots(col = "status",
          alpha = 0.6,
          palette = "YlOrRd") +
  tm_view(set.zoom.limits = c(9, 12))

prob_T_refined <- tm_shape(Osun) +
  tm_polygons(alpha = 0.4) +
  tm_shape(gwr_sf.fixed_refined) + 
  tm_dots(col = "yhat",
          border.col = "gray60",
          border.lwd = 1) +
  tm_view(set.zoom.limits = c(9, 12))

tmap_arrange(actual, prob_T_refined, 
             asp = 1, ncol = 2, sync = TRUE)

We see that the predictions are largely aligned with the actual status of the water points, in line with the 88% accuracy rate.